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To develop the acoustic wave equation, fluid flow and its viscosity are neglected in the fluid medium. We will develop the velocity potential formulation for the acoustic wave equation. 228) ∂t   in which ρ is the fluid density, v is the velocity vector, ∇ is the gradient operator, and t denotes time. 229) where s is called condensation, and ρo is an ambient fluid density. 231) ∂t Here, p is the pressure. 234) where B is the bulk modulus. 236) ∂t 2 where c is the speed of sound, and c2 = B/ρo.

68) ∂t 2 Here, v is the transverse displacement, ρ is the mass density, and A is the cross-sectional area. 5 Free-body diagram of an infinitesimal beam. 73) ∂x 2 where EI is the beam rigidity. 74) 2 ∂t ∂x if EI is constant. This is the governing equation for classical beam theory. 75) ∂t ∂x   This is the strong formulation. 77) 0 ∂x  0 e in which n is the number of elements. For the beam element, we need shape functions. 32 cannot be used for the beam bending. 32 provide a continuous solution from one element to the next element, but their derivative is not continuous between two neighboring elements.

However, the shape functions are expressed in terms of ξ and η. As a result, we need to relate the two coordinate systems using mathematical mapping. 14. Likewise, the inside and boundary of the square element in the natural coordinate are mapped to the inside and boundary of the quadrilateral in the physical domain. Let us compute the Jacobian matrix associated with the mapping. 184) i i i=1 Now, let us discuss how to take derivatives of the shape functions Hi(ξ, η) in terms of x and y. To this end, we consider functions φ(x, y) and (x, y) are functions of (ξ, η).

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